Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Lindelöf property and fragmentability

Authors: B. Cascales, I. Namioka and G. Vera
Journal: Proc. Amer. Math. Soc. 128 (2000), 3301-3309
MSC (2000): Primary 46A50, 46B22; Secondary 54C35
Published electronically: April 28, 2000
MathSciNet review: 1695167
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $K$ be a compact Hausdorff space and $C(K)$ the space of continuous real functions on $K$. In this paper we prove that any $t_{p}(K)$-Lindelöf subset of $C(K)$ which is compact for the topology $t_{p}(D)$ of pointwise convergence on a dense subset $D\subset K$ is norm fragmented; i.e., each non-empty subset of it contains a non-empty $t_{p}(D)$-relatively open subset of small supremum norm diameter. Several applications are given.

References [Enhancements On Off] (What's this?)

  • 1. A. V. Arkhangleskii, Problems in $C_p$-theory, Open problems in topology, 601-615, North-Holland, Amsterdam, 1990.
  • 2. J. Bourgain, D. Fremlin, and M. Talagrand, Pointwise compact sets of Baire measurable functions, Amer. J. Math., 100 (1978), 845-886. MR 80b:54017
  • 3. R. D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodým property, LNM, Springer-Verlag, 993, 1983. MR 85d:46023
  • 4. B. Cascales, G. Manjabacas, and G. Vera, Fragmentability and compactness in $C(K)$-spaces, Studia Mathematica, 131 (1998), 73-87. MR 99d:46038
  • 5. B. Cascales and G. Vera, Topologies weaker than the weak topology of a Banach space, J. Math. Anal. Appl., 182 (1994), 41-68. MR 95c:46017
  • 6. G. Debs, Points de continuité d'une fonction séparément continue. II, Proc. Amer. Math. Soc., 99 (1987), 777-782. MR 88d:54013
  • 7. R. Deville, Parties faiblement de Baire dans les espaces de Banach. Applictions à la dentabilité et à l'unicité de certains préduaux, C. R. Acad. Sc. Paris. Serie I, 298 (1984), 129-131. MR 85h:46029
  • 8. J. E. Jayne and C. A. Rogers, Borel selectors for upper semicontinuous set-valued maps, Acta Math, 155 (1985), 41-79. MR 87a:28011
  • 9. W. Moran, Separate continuity and support of measures, J. London Math. Soc., 44 (1969), 320-324. MR 38:4642
  • 10. I. Namioka, Separate continuity and joint continuity, Pac. J. Math., 51 (1974), 515-531. MR 51:6693
  • 11. I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika, 34 (1989), 258-281. MR 89i:46021
  • 12. R. Pol, On pointwise and weak topology in function spaces, Preprint Nr. 4/84. Warszawa, 1984.
  • 13. H. P. Rosenthal, A characterization of Banach spaces containing $\ell ^1$, Proc. Nat. Acad. Sci. U.S.A., 71 (1974), 2411-2413. MR 50:10773
  • 14. W. Sierpinski, Sur une suite infinie de fonctions de classe 1 dont toute fonction d'acumulation est non mesurable, Fund. Math., 33 (1945), 104-105. MR 8:18a
  • 15. V. V. Srivatsa, Baire class 1 selectors for upper semi-continuous set-valued maps, Trans. Amer. Math. Soc. 337 (1993), 609-624. MR 93h:54013
  • 16. C. Stegall, Functions of the first Baire class, Proc. Amer. Math. Soc., 111 (1991), 981-991. MR 91k:26003
  • 17. M. Talagrand, Deux generalisations d'un théorème de I. Namioka, Pacific J. Math., 81 (1979), 239-251. MR 80k:54018
  • 18. G. Vera, Baire measurability of separately continuous functions, Quart. J. Math. Oxford, (2), 39 (1988), 109-116. MR 89e:28007

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46A50, 46B22, 54C35

Retrieve articles in all journals with MSC (2000): 46A50, 46B22, 54C35

Additional Information

B. Cascales
Affiliation: Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain

I. Namioka
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195–4350

G. Vera
Affiliation: Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain

Keywords: Pointwise compactness, Radon-Nikod\'ym compact spaces, fragmentability
Received by editor(s): July 20, 1998
Received by editor(s) in revised form: December 20, 1998
Published electronically: April 28, 2000
Additional Notes: The first and third authors were partially supported by research grant DGES PB 95–1025.
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society